It's been a while, but I wanted to give another update, which will hopefully give people some ideas to try if they find themselves in a similar situation with their bike.
So since 2 years ago, my GSXR750 started getting more and more difficult to start again. Based on what DownUnder commented, I thought maybe there were some old, worn out wires. And I do think it's part of it. While cutting the relay plugs and hard-wiring them directly didn't seem to make any noticeable change, going back and cleaning up some of the other wires, spraying with contact cleaner and covering with dielectric grease did seem to help, but it was still taking a few tries to start the bike each time.
Since the bike was occasionally able to fire up, I assumed there should be fuel and air, even if they're not perfect. Sparks were still good. Battery still tested at the right voltage with the bike off/cranking/running. Alternator and rectifier were also still good.
So I started thinking about the starter motor again. It spun, but I wondered if it might be spinning too slow. So I took apart the motor thinking maybe it needs new brushes. Instead, I found one of the springs for the brushes was a little corroded and sticking. I cleaned it up a abit and used some WD-40 (not sure if there's something better I should be using inside an electric motor) so it was moving smoothly. Sanded the contacts a bit, cleaned everything up and put everything back together. I tried spinning the shaft by hand and found it seizing, so I realized I put the spacers back incorrectly. After taking it apart, putting the spacers in correctly and reassembling, this time the shaft was able to spin pretty easily. I put the motor back in my bike, charged my battery back up to full. So far, the bike has been starting up pretty much right away each time.
Now the only thing left is that the idle RPM's fluctuate a bit. I've found at least one source of an air leak, and will be trying to see if there's any others. If after fixing all the air leaks up and there's still some fluctuation, I'll have to see if there's an easy way to test my fuel injectors. But for now, I have my idle knob adjusted to allow me to ride without much issue.